Spinon magnetic resonance of quantum spin liquids

Zhu-Xi Luo,1 Ethan Lake,1 Jia-Wei Mei,2 and Oleg A. Starykh1

1Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA2Department of Materials Science and Engineering,

University of Utah, Salt Lake City, Utah 84112, USA(Dated: January 23, 2018)

We describe electron spin resonance in a quantum spin liquid with significant spin-orbit coupling.We find that the resonance directly probes spinon continuum which makes it an efficient and in-formative probe of exotic excitations of the spin liquid. Specifically, we consider spinon resonanceof three different spinon mean-field Hamiltonians, obtained with the help of projective symmetrygroup analysis, which model a putative quantum spin liquid state of the triangular rare-earth anti-ferromagnet YbMgGaO4. The band of absorption is found to be very broad and exhibit strong vanHove singularities of single spinon spectrum as well as pronounced polarization dependence.

INTRODUCTION

Electron spin resonance (ESR) and its variants in mag-netically ordered systems - ferromagnetic and antiferro-magnetic resonances - represent one of the most preciseand frequently used spectroscopic probes of excitationsof magnetic media. The essence of the magnetic reso-nance technique consists in measuring absorption of elec-tromagnetic radiation (usually in the microwave rangeof frequencies) by a sample material which is (typically)subjected to an external static magnetic field. The ab-sorption is caused by coupling of magnetic degrees of free-dom to the magnetic field of the electromagnetic wave.Given very large wavelength of the microwave, the ESRabsorption is driven by zero wavevector (q = 0, or verti-cal) transitions between states with different Sz projec-tions of magnetic dipole moment on the direction per-pendicular to the magnetic field of the EM radiation.

In a spin system with isotropic exchange, the absorp-tion spectrum of an AC magnetic field is a δ-functionpeak at the frequency equal to that of the Zeeman en-ergy, independently of the exchange interaction strength.This is a consequence of the fact that at q = 0 EM radia-tion couples to the total magnetic moment, which for anSU(2) invariant system commutes with the Hamiltonian[1]. Therefore, any deviation of the absorption spectrumfrom the δ-function shape implies violation of the spin-rotation symmetry, caused either by anisotropic termsin the Hamiltonian (explicit symmetry breaking) or bythe development of long-range magnetic order below thecritical temperature (spontaneous symmetry breaking).This is the key reason for ESR’s utility.

The goal of our work is to explore applications of ESRto highly entangled phase of magnetic matter - the quan-tum spin liquid (QSL) [2]. This intriguing novel quan-tum state manifests itself via non-local elementary exci-tations - spinons - which behave as fractions of ordinaryspin waves. Local spin operator becomes a compositeof two or more spinons, which immediately implies thatdynamic spin susceptibility measures multi-spinon con-

tinuum. In principle, the best probe of the spinon con-tinuum is provided by inelastic neutron scattering whichprobes spinons at finite wave vector q and frequency ω.By now several textbook-quality experiments have pro-vided us with unambiguous signatures of multi-particlecontinua [3–5]. In practice, however, such state of theart measurements require large high-quality single crys-tals which quite frequently are not available.

We posit here that ESR, with its exceptionally high en-ergy resolution, represents an appealing complimentaryspectroscopic probe of spinons - spinon magnetic reso-nance (SMR). The key requirement for turning it intoa full-fledged probe of spinon dynamics consists in theabsence of spin-rotational invariance. This requirementstems from the mentioned above ‘insensitivity’ of ESRto the details of excitations spectra in SU(2) invariantmagnetic materials. Note that the SU(2) invariance is,at best, a theoretical approximation to the real world ma-terials which always suffer from some kind of magneticanisotropy.

Moreover, over the past fifteen years the field of QSLhas evolved dramatically away from the spin-rotationalinvariance requirement explicit in many foundational pa-pers [6–8]. The absence of spin-rotational invariance hasevolved from the ‘real world’ annoyance to the virtue[2, 9, 10]. Indeed, the first and still the most direct andunambiguous demonstration of the gapless QSL phasecame from Kitaev’s exact solution of the fully anisotropichoneycomb lattice model [11] which does not conserve to-tal spin.

Importantly, a large number of very interesting andnot yet understood materials, such as α-RuCl3 [12],YbMgGaO4 [13, 14], Yb2Ti2O7 [15, 16] and many otherpyrochlores [17], and even organic BEDT-TTF andBEDT-TSF salts[18], showing promising QSL-like fea-tures are known to possess significant spin-orbit interac-tion and are described by spin Hamiltonians with signif-icant asymmetric exchange and pseudo-dipolar terms. Itis precisely this class of low-symmetry spin models wefocus on in the present study.

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We illustrate our idea by considering spin-liquid stateproposed to describe a spin-orbit-coupled triangular lat-tice Mott insulator YbMgGaO4. The appropriate spinHamiltonian has been argued to be that of XXZ modelwith interactions between nearest (with J ∼ 1K) andnext-nearest neighbors on the triangular lattice togetherwith a pseudo-dipolar term [19–21], of J±± kind in no-tations of [15], between nearest neighbors (J±± ∼ 0.2K).Most recently, polarized neutron scattering data were in-terpreted in favor of significant Jz± interaction [22]. ThisHamiltonian does not conserve total spin Stot.

Inelastic neutron scattering experiments reveal broadspin excitations continuum [23, 24], consistent with frac-tionalized QSL with spinon Fermi surface. At the sametime experimental evidence of significant disorder effects[21, 24–26], capable of masking ‘pristine’ physics of thematerial, is mounting.

Our goal here is to add to the ongoing discussion on thenature of the ground state of YbMgGaO4 by pointing outthat ESR can serve as a very useful probe of QSL withsignificant built-in spin-orbit interactions. We there-fore accept spin-liquid hypothesis and focus on fermionicU(1) symmetric spin-liquid ground states, proposed forthis material previously [23, 27]. We rely on the well-established projective symmetry group (PSG) analysisof possible U(1) spin liquids [27–31]. The spin-orbitalnature of the effective spin-1/2 local moment of Yb3+

ion implies that under the space group symmetry oper-ations both the direction and the position of the localspin are transformed. The symmetry operations includetranslations T1,2 along the major axis a1,2 of the crystallattice, a rotation C2 by π around the in-plane vectora1 + a2, a counterclockwise rotation C3 by 2π/3 aroundthe lattice site, and the (three-dimensional) inversion Iabout the lattice site. Following [27], it is convenientto combine C3 and I operations into a composite oneC6 ≡ C−1

3 I. (Note that the original C6 lattice rotationby 2π/6 around the lattice site is not the symmetry ofYbMgGaO4 due to alternating - above and below theplane - location of oxygens at the centers of consecutiveelementary triangles [13].)

These symmetries strongly constrain possible U(1)mean-field spinon Hamiltonians and result in 8 differentPSG states, of U1A and U1B kind. U1A states maintainperiodicity of the original lattice and their band struc-ture consists of just two spinon bands. U1B states areπ-flux states with doubled unit cell. Equivalently, theirband structure contains 4 spinon bands. For the sakeof simplicity, we focus on the U1A family in the follow-ing (description of U1B increases algebraic complexitywithout adding any new essential physics). The U(1)mean-field spinon Hamiltonian is parameterized by sev-eral hopping amplitudes - t1,2 describe spin-conservinghopping between the nearest and the next-nearest neigh-bors and t′1,2 describes analogous non-spin-conservinghops. PSG analysis fixes relative phases between hopping

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0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

ω/t1

2I(ω

)/h2

K M KΓ Γ

U1A11

-2

-1

1

2

FIG. 1. (Color online) Plot of 2I(ω)/|h|2 vs ω/t1 for differ-ent polarizations θ = 0 (blue dots), π/4 (orange squares),and π/2 (green rhombi) for U1A11 state. The insert showsspinon band structure along the high-symmetry path Γ-K-M-K-Γ in the Brillouin zone. Vertical red line illustrates opticaltransitions between spinon bands.

amplitudes on the bonds related by the space group oper-ations (see Supplement [32] for details of the derivation).The magnitudes of these hoppings are not determinedby PSG. This requires a separate variational calculationof the ground state energy which is not attempted here.We do expect, on physical grounds, that for the spinmodel with predominant isotropic nearest-neighbor spinexchange and subleading asymmetric J±± terms, the fol-lowing estimate should hold t1 > t′1 > t2 > t′2.

There are four mean-field Hamiltonians in U1A family,labeled by U1AnC2

nC6(nC2

, nC6∈ {0, 1}).They have the

simple form

H =∑k

(f†k↑, f†k↓)

(ωk + εk ηkη∗k ωk − εk

)(fk↑fk↓

), (1)

where k-dependent ωk, εk, ηk are listed in [32]. Spin-orbitinteraction appears via spin-non-conserving hopping ηkin (1). The U1A00 state is characterized by finite ωk andzero εk and ηk, while U1A01 and U1A11 have ωk = 0 andfinite εk and ηk. In the calculations below we set t1 = 1and t′1 = 0.8, t′2 = 0.3 for U1A11, while t1 = 0 for U1A01and we choose t′1 = 1, t2 = 0.8t′1, t

′2 = 0.4t′1 for it. U1A10

turns out to be non-physical since its Hamiltonian matrixis zero, t1,2 = t′1,2 = 0. ‘Accidental’ nature of U1A00state is manifested by the absence of any spin-dependenthopping in its Hamiltonian - this state happens to bemore symmetric than the spin Hamiltonian it describesand is characterized by the large Fermi surface [23].

We focus on most physically relevant U1A01 andU1A11 states, for which ωk = 0. The resulting fermionbands are easy to find, Eν=1,2(k) = (−1)νE(k) =

(−1)ν√ε2k + |ηk|2. U1A11 state possess symmetry-

protected Dirac nodes at Γ and M points of the hexagonalBrillouin zone, while U1A01 has additional Dirac nodesat K points as well.

3

Interaction with monochromatic radiation linearly po-larized along direction n = (sin θ cosφ, sin θ sinφ, cos θ)is described by V (t) = −h(t) · Stot, i.e.

V (t) = he−iωtn · 1

2

∑r

(f†r↑, f†r↓)σ

(fr↑fr↓

)(2)

Within linear response theory the rate of energy absorp-tion I(ω) = −ωχ′′nn(ω)|h|2/2 is determined by the imag-inary part of q = 0 Fourier transform of the dynamicsusceptibility [1] χnn(t, r) = −iΘ(t)〈[Sr(t) · n,S0(0) · n]〉,with Θ being the Heaviside function. Straightforwardcalculation gives

χnn(ω) =1

4N

∑k

nkα − nkβω + Eα(k)− Eβ(k) + i0

×(U+k σ

aUk)αβ(U+k σ

bUk)βαnanb (3)

Here nkα is the occupation number of the band α, Uk isunitary diagonalizing matrix connecting spinor of origi-nal fermions to that of the band ones, fk,α = (Uk)αβbk,β ,and summation over repeated indices is implied. Eq.(3)shows that in the spin-degenerate U1A00 state, for whichnkα = nkβ , the susceptibility is strictly zero. Therefore,in agreement with general discussion above, no energyabsorption occurs in the absence of external magneticfield for this state. The condition nkα ≈ nkβ is also satis-fied at high temperature of the order of spinon bandwidth(which is of the order of exchange J) when spinon reso-nance disappears. We therefore expect the width of theresonance to increase when the temperature is lowered.It is worth noting that the lowest temperature of ESRstudy [14] is 1.8K, which makes it a high-temperaturemeasurement.

At zero temperature absorption at frequency ω is possi-ble only via vertical transitions from the filled lower band(α = 1, nk1 = 1) to the empty upper one (β = 2, nk2 = 0)and therefore

χ′′nn(ω) = − π

4N

∑k

δ(ω − 2E(k))(U+k σ

aUk)12

×(U+k σ

bUk)21nanb (4)

After some algebra, the product of matrix elements 12and 21 of the rotated Pauli matrices in the equationabove simplifies to

χ′′nn(ω) = − π

4N

∑k

δ(ω − 2E(k))

E(k)2

[(ε2k + η′′2k ) sin2 θ cos2 φ

+(ε2k + η′2k ) sin2 θ sin2 φ+ |ηk|2 cos2 θ]. (5)

It can be shown [32] that the omitted off-diagonalterms, containing products nxny, nxnz and nynz, are allzero. Moreover, terms proportional to cos2 φ and sin2 φare actually equal, so that the absorption only dependson the azimuthal angle θ with respect to the normal tothe magnetic layer.

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◆

0 1 2 3 4 50.0

0.2

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0.6

0.8

1.0

1.2

ω/t1

2I(ω

)/h2 K M KΓ Γ

U1A01

-2-1

12

FIG. 2. (Color online) Plot of 2I(ω)/|h|2 vs ω/t′1 for differentpolarizations θ = 0 (blue dots), π/4 (orange squares), andπ/2 (green rhombi) for U1A01 state. The insert shows spinondispersion along the path in Fig. 1.

Two features of this result are worth noting. First, theabsorption takes place over the wide band of frequencies,min(E) = 0 < ω/2 < max(E), which covers the fullbandwidth of two-spinon continuum. Second, Eq.(5) de-scribes zero-field absorption, which does not require anyexternal static magnetic field B. Both of these are di-rect consequence of the absence of spin conservation inEq.(1).

U1A11 state: Figure 1 shows scaled absorption in-tensity, 2I(ω)/|h|2 = −ωχ′′nn(ω), for different polariza-tions. Polarization dependence is strong. The plot isobtained by numerical integration of (5), with frequencysteps of ∆ω = 0.05, over the primitive cell of the re-ciprocal lattice (k = (k1, k2), where k1,2 ∈ (0, 2π), see[32] for details). We approximate the delta-function bythe Lorenzian δ(x) ≈ π−1d/(d2 + x2) with d = 0.01.We checked that d = 0.05 results in the same out-come. As expected, and also easy to check analyti-cally, χ′′nn(ω) ∼ ω at small frequencies. This is theconsequence of Dirac nodes at Γ and M points. Behav-ior near the upper boundary, ω ≈ 3

√3, is determined

by the vicinity of K point where ε(K) = const whileη(K) = 0. As a result, one obtains χ′′zz ∼ 3

√3 − ω,

while at θ = π/2 susceptibility terminates discontinu-ously in a step-like fashion, χ′′nn = χ′′xx ∼ Θ(3

√3 − ω).

The rounding of the step-function behavior in Fig. 1,for θ 6= 0, is caused by the finite width of numericaldelta-function used in the integration over the Brillouinzone. The peak in the middle of the absorption band,at ω ≈ 2.43, is caused by the van Hove singularity, ofthe saddle point kind, of E(k) at k0 = (k0, 2π − k0)and symmetry-related points. Here k0 ≈ 1.97 andE(k ≈ k0) ≈ 1.215 + 1.31(k1 + k2)2 − 0.23(k1 − k2)2.The saddle-point produces logarithmically divergent con-tribution, χ′′nn ∼ ln |ω − 2.43|, which matches numericaldata in Fig. 1 perfectly.

U1A01 state: SMR of this phase is shown in Figure 2.

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◆

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0.0

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ω/t1

2I(ω

)/h2

FIG. 3. (Color online) Plot of 2I(ω)/|h|2 vs ω/(2t1) for po-larizations θ = 0 (blue dots), π/4 (orange squares), and π/2(green rhombi) for U1A11 state in the presence of magneticfield Bz = 1 . Note the appearance of strong van Hove singu-larity at ω ≈ 4.2t1. Thin red line shows Zeeman response ofU1A00 state.

It is seen to host two van Hove singularities which can bequalitatively understood as a direct consequence of theadditional, in comparison with U1A11 state, Dirac conein the spinon dispersion at the K point. The presenceof the symmetry-protected node at the K point resultsin a stronger variation of spinon dispersion in the Bril-louin zone and causes the appearance of additional saddlepoints.

U1A11 state in magnetic field: external magneticfield adds further variations to the spinon absorption in-tensity. We illustrate this with the case of U1A11 statesubject to magnetic field B = Bz z along the normal tothe magnetic layer. It should be noted that PSG anal-ysis underlaying our consideration assumes time-reversal(TR) symmetry. Therefore we treat magnetic field per-turbatively, by coupling it to the local TR-odd combi-nation of spinons which is just BzS

zr ∼ Bzf

†rασ

zαβfrβ .

Thus magnetic field enters (1) via εk → εk − Bz/2 andgaps out Dirac nodes. The minimal excitation energy be-comes min(E) = Bz/2 and absorption intensity acquiresthreshold behavior I(ω) ∼ Θ(ω − Bz). This behavior isillustrated in Figure 3, which also shows development ofadditional spectral features at ω ≈ 4.2, see [32]. In-planemagnetic field lowers symmetry of the spin Hamiltonianfurther and its consideration is left for future studies.

This unusual response should be contrasted with thatof the large-Fermi-surface state U1A00. Here B = Bz zleads to the Zeeman splitting of spinon up- and down-spinbands Eν = ωk ∓ Bz/2 and therefore, according to (3),one finds standard result for magnetically isotropic mediaχ′′nn(ω) ∼ sin2 θ δ(ω − Bz). This is consistent with theearlier analysis of [33], where a weak magnetic field B =Bz z was added to the mean-field Hamiltonian similarly.Off the Γ point, i.e. for q 6= 0, one finds broad continuumcorresponding to the spinon particle-hole excitations [33].

Discussion: Physical arguments leading to Eq.(5)

are very general and rely on absence of long-range mag-netic order, existence of fractionalized elementary excita-tions, which ensure a continuum-like response to exter-nal probes, and significant built-in spin-orbit interaction,which leads to non-conservation of spin and makes zero-field absorption possible in a wide range of frequencies.All of these are very generic conditions which are sat-isfied by essentially every model of spin liquids of U(1)and Z2 type (but not by spin-conserving SU(2) ones).The restriction to low symmetry spin liquids is not reallya handicap as it turned out that the number of possi-ble spin liquids with reduced U(1) and Z2 vastly out-numbers that of SU(2) symmetric ones [30, 34, 35]. Inparticular, the SMR should be present in the celebratedKitaev’s honeycomb model [11], as was emphasized indynamic structure calculations of [36–39]. There too onecan see anisotropic spin structure factor Saa(q = 0, ω),with Szz 6= Sxx/yy, and sharp van Hove singularities inthe Majorana fermion density of states. The similarity isnot accidental - it follows from the linear mapping be-tween Majorana and projective spinon representations[40, 41]. Unlike the situation described here, in the ex-actly solvable gapless Abelian region dynamic responseappears above a finite threshold energy (which is the en-ergy cost of creating Z2 fluxes). However generic spinexchange perturbations turn the response gapless [42],so that Saa(q = 0, ω) ∼ ω at low energy. Resonant in-elastic x-ray (RIXS), Raman scattering and parametricpumping of the Z2 Kitaev spin liquid results in a gaplessand extended in energy continuum too [43–46].

Our theory can be broadly thought of as an extensionof one-dimensional theories of ESR in spin chains withDzyaloshinskii-Moriya interactions [1, 47–49]. In one di-mension, fractionalized nature of spinons is very well es-tablished and theories based on them describe ESR ex-periments exceedingly well, both in gapless [50–52] andgapped [53, 54] settings.

Another important connection is provided by electricdipole spin resonance (EDSR) which describes absorp-tion of EM radiation in conductors with pronounced spin-orbit interaction which mediates coupling of AC electricfield to the electron spin [55]. Here, spin-rotational asym-metry causes strong absorption which is controlled by thereal part of optical conductivity [56–62].

Somewhat surprisingly, energy absorption due to cou-pling of spins to AC electric field is also possible in strongMott insulators, provided they are built of frustrated tri-angular units, in which virtual charge fluctuations pro-duce spin-dependent electric polarization [63, 64]. Hintsof this physics were recently observed in herbertsmithiteand α-RuCl3 antiferromagnets [65–67].

Simple calculations of SMR presented here are basedon mean-field spinon Hamiltonians derived with the helpof PSG formalism. They do not include gauge fluctua-tions which undoubtedly are present in the theory. Thesefluctuations are certain to affect exponents characterizing

5

sharp features of χ′′nn(ω), such as for example behaviornear the van Hove singularity and/or near lower/upperedge of the two-spinon continuum. (Disorder, in the formof Mg/Ga mixing, leads to distribution of g-factors [25]which also broadens magnetic response.) In addition, byanalogy with critical Heisenberg chain [68], we expectfour-spinon contributions to the susceptibility to affectthe high-frequency behavior. However, these importanteffects can not reduce spinon absorption bandwidth andeliminate other outstanding features of the SMR foundhere. It should also be noted that SMR is not specificto fermionic spinons and indeed extension of the theoryto bosonic PSG is possible as well [69, 70]. We there-fore conclude that spinon magnetic resonance representsan efficient and informative probe of exotic excitations ofspin-orbit-coupled quantum spin liquids.

O.A.S. thanks Leon Balents for extensive and insight-ful discussions of YbMgGaO4 and PSG formalism, MikeHermele for informative remarks, Sasha Chernyshev fornumerous fruitful conversations, Natasha Perkins andDima Pesin for comments on the manuscript. Z.-X. L.thanks Yao-Dong Li for helpful discussions. O.A.S. issupported by the National Science Foundation grant NSFDMR-1507054.

Notes Added. Manuscript [71], which appeared afterour submission, contains detailed comparison of groundstate energies of various U(1) PSG states.

[1] M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410(2002).

[2] L. Savary and L. Balents, Reports on Progress in Physics80, 016502 (2017).

[3] D. C. Dender, P. R. Hammar, D. H. Reich, C. Broholm,and G. Aeppli, Phys. Rev. Lett. 79, 1750 (1997).

[4] R. Coldea, D. A. Tennant, and Z. Tylczynski, Phys. Rev.B 68, 134424 (2003).

[5] B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler,Nat Mater 4, 329 (2005).

[6] P. W. Anderson, Science 235, 1196 (1987).[7] I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774

(1988).[8] N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773

(1991).[9] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba-

lents, Annual Review of Condensed Matter Physics 5, 57(2014), http://dx.doi.org/10.1146/annurev-conmatphys-020911-125138.

[10] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, An-nual Review of Condensed Matter Physics 7, 195(2016), http://dx.doi.org/10.1146/annurev-conmatphys-031115-011319.

[11] A. Kitaev, Annals of Physics 321, 2 (2006).[12] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li,

M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu,J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moess-ner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler,Nat Mater 15, 733 (2016).

[13] Y. Li, H. Liao, Z. Zhang, S. Li, F. Jin, L. Ling, L. Zhang,Y. Zou, L. Pi, Z. Yang, J. Wang, Z. Wu, and Q. Zhang,Scientific Reports 5, 16419 (2015).

[14] Y. Li, G. Chen, W. Tong, L. Pi, J. Liu, Z. Yang, X. Wang,and Q. Zhang, Phys. Rev. Lett. 115, 167203 (2015).

[15] K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents,Phys. Rev. X 1, 021002 (2011).

[16] L. Pan, S. K. Kim, A. Ghosh, C. M. Morris, K. A. Ross,E. Kermarrec, B. D. Gaulin, S. M. Koohpayeh, O. Tch-ernyshyov, and N. P. Armitage, Nat. Commun. 5, 4970(2014).

[17] M. J. P. Gingras and P. A. McClarty, Reports on Progressin Physics 77, 056501 (2014).

[18] S. M. Winter, K. Riedl, and R. Valentı, Phys. Rev. B95, 060404 (2017).

[19] Y.-D. Li, X. Wang, and G. Chen, Phys. Rev. B 94,035107 (2016).

[20] Y.-D. Li, Y. Shen, Y. Li, J. Zhao, and G. Chen, ArXive-prints (2016), arXiv:1608.06445 [cond-mat.str-el].

[21] Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Cherny-shev, ArXiv e-prints (2017), arXiv:1703.02971 [cond-mat.str-el].

[22] S. Toth, K. Rolfs, A. R. Wildes, and C. Ruegg, ArXive-prints (2017), arXiv:1705.05699 [cond-mat.str-el].

[23] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan,Q. Wang, H. C. Walker, P. Steffens, M. Boehm, Y. Hao,D. L. Quintero-Castro, L. W. Harriger, M. D. Frontzek,L. Hao, S. Meng, Q. Zhang, G. Chen, and J. Zhao, Na-ture 540, 559 (2016).

[24] J. A. M. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu,M. B. Stone, H. Zhou, and M. Mourigal, Nat Phys 13,117 (2017).

[25] Y. Li, D. Adroja, R. I. Bewley, D. Voneshen, A. A. Tsir-lin, P. Gegenwart, and Q. Zhang, Phys. Rev. Lett. 118,107202 (2017).

[26] Y. Xu, J. Zhang, Y. S. Li, Y. J. Yu, X. C. Hong, Q. M.Zhang, and S. Y. Li, Phys. Rev. Lett. 117, 267202(2016).

[27] Y.-D. Li, Y.-M. Lu, and G. Chen, Phys. Rev. B 96,054445 (2017).

[28] X.-G. Wen, Phys. Rev. B 65, 165113 (2002).[29] X. G. Wen, Quantum Field Theory of Many-Body Sys-

tems: From the Origin of Sound to an Origin of Light andElectrons, Oxford Graduate Texts (OUP Oxford, 2004).

[30] J. Reuther, S.-P. Lee, and J. Alicea, Phys. Rev. B 90,174417 (2014).

[31] S. Bieri, C. Lhuillier, and L. Messio, Phys. Rev. B 93,094437 (2016).

[32] Supplemental Material reference.[33] Y.-D. Li and G. Chen, ArXiv e-prints (2017),

arXiv:1703.01876 [cond-mat.str-el].[34] T. Dodds, S. Bhattacharjee, and Y. B. Kim, Phys. Rev.

B 88, 224413 (2013).[35] B. Huang, Y. B. Kim, and Y.-M. Lu, Phys. Rev. B 95,

054404 (2017).[36] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-

ner, Phys. Rev. Lett. 112, 207203 (2014).[37] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess-

ner, Phys. Rev. B 92, 115127 (2015).[38] K. O’Brien, M. Hermanns, and S. Trebst, Phys. Rev. B

93, 085101 (2016).[39] A. Smith, J. Knolle, D. L. Kovrizhin, J. T. Chalker, and

R. Moessner, Phys. Rev. B 93, 235146 (2016).[40] F. J. Burnell and C. Nayak, Phys. Rev. B 84, 125125

6

(2011).[41] Y.-Z. You, I. Kimchi, and A. Vishwanath, Phys. Rev. B

86, 085145 (2012).[42] X.-Y. Song, Y.-Z. You, and L. Balents, Phys. Rev. Lett.

117, 037209 (2016).[43] G. B. Halasz, N. B. Perkins, and J. van den Brink, Phys.

Rev. Lett. 117, 127203 (2016).[44] J. Knolle, G.-W. Chern, D. L. Kovrizhin, R. Moessner,

and N. B. Perkins, Phys. Rev. Lett. 113, 187201 (2014).[45] G. B. Halasz, B. Perreault, and N. B. Perkins, ArXiv

e-prints (2017), arXiv:1705.05894 [cond-mat.str-el].[46] A. A. Zvyagin, Phys. Rev. B 95, 064428 (2017).[47] M. Oshikawa and I. Affleck, Phys. Rev. Lett. 82, 5136

(1999).[48] S. Gangadharaiah, J. Sun, and O. A. Starykh, Phys.

Rev. B 78, 054436 (2008).[49] H. Karimi and I. Affleck, Phys. Rev. B 84, 174420 (2011).[50] S. A. Zvyagin, A. K. Kolezhuk, J. Krzystek, and R. Fey-

erherm, Phys. Rev. Lett. 95, 017207 (2005).[51] K. Y. Povarov, A. I. Smirnov, O. A. Starykh, S. V.

Petrov, and A. Y. Shapiro, Phys. Rev. Lett. 107, 037204(2011).

[52] M. Halg, W. E. A. Lorenz, K. Y. Povarov, M. Mansson,Y. Skourski, and A. Zheludev, Phys. Rev. B 90, 174413(2014).

[53] V. N. Glazkov, M. Fayzullin, Y. Krasnikova, G. Skoblin,D. Schmidiger, S. Muhlbauer, and A. Zheludev, Phys.Rev. B 92, 184403 (2015).

[54] M. Ozerov, M. Maksymenko, J. Wosnitza, A. Honecker,C. P. Landee, M. M. Turnbull, S. C. Furuya, T. Gia-marchi, and S. A. Zvyagin, Phys. Rev. B 92, 241113(2015).

[55] E. I. Rashba, Soviet Physics Uspekhi 7, 823 (1965).[56] A.-K. Farid and E. G. Mishchenko, Phys. Rev. Lett. 97,

096604 (2006).[57] A. Abanov, V. L. Pokrovsky, W. M. Saslow, and P. Zhou,

Phys. Rev. B 85, 085311 (2012).[58] R. Glenn, O. A. Starykh, and M. E. Raikh, Phys. Rev.

B 86, 024423 (2012).[59] C. Sun and V. L. Pokrovsky, Phys. Rev. B 91, 161305

(2015).[60] S. Maiti, M. Imran, and D. L. Maslov, Phys. Rev. B 93,

045134 (2016).[61] V. L. Pokrovsky, Low Temperature Physics 43, 211

(2017), http://dx.doi.org/10.1063/1.4976632.[62] A. Bolens, H. Katsura, M. Ogata, and S. Miyashita,

ArXiv e-prints (2017), arXiv:1704.03153 [cond-mat.str-el].

[63] L. N. Bulaevskii, C. D. Batista, M. V. Mostovoy, andD. I. Khomskii, Phys. Rev. B 78, 024402 (2008).

[64] A. C. Potter, T. Senthil, and P. A. Lee, Phys. Rev. B87, 245106 (2013).

[65] D. V. Pilon, C. H. Lui, T. H. Han, D. Shrekenhamer,A. J. Frenzel, W. J. Padilla, Y. S. Lee, and N. Gedik,Phys. Rev. Lett. 111, 127401 (2013).

[66] N. J. Laurita, G. G. Marcus, B. A. Trump, J. Kinder-vater, M. B. Stone, T. M. McQueen, C. L. Broholm, andN. P. Armitage, ArXiv e-prints (2017), arXiv:1704.04228[cond-mat.str-el].

[67] A. Little, L. Wu, P. Lampen-Kelley, A. Banerjee, S. Pan-tankar, D. Rees, C. A. Bridges, J.-Q. Yan, D. Mandrus,S. E. Nagler, and J. Orenstein, ArXiv e-prints (2017),arXiv:1704.07357 [cond-mat.str-el].

[68] J.-S. Caux and R. Hagemans, Journal of Statistical Me-

chanics: Theory and Experiment 2006, P12013 (2006).[69] F. Wang and A. Vishwanath, Phys. Rev. B 74, 174423

(2006).[70] L. Messio, C. Lhuillier, and G. Misguich, Phys. Rev. B

87, 125127 (2013).[71] J. Iaconis, C. Liu, G. B. Halasz, and L. Balents, ArXiv

e-prints (2017), arXiv:1708.07856 [cond-mat.str-el].

7

Supplementary Information for “ Spinon magnetic resonance of quantum spin liquids”Zhu-Xi Luo, Ethan Lake, Jia-Wei Mei, and Oleg A. Starykh

The supplementary material is arranged as follows: Part A discusses geometry of the lattice, PSG analysis is reviewedin Part B, spectra for U1A11 and U1A01 are presented in Part C, and the discussion of dynamical susceptibility canbe found in Part D.

A. THE LATTICE

This section introduces the coordinate systems in both real space and Fourier space.

A1. Real Space

The rare-earth Yb atoms in YbMgGaO4 form a triangular lattice. We use the following intralayer coordinatesr = xa1 + ya2, with

a1 = (1, 0), a2 =

(−1

2,

√3

2

). (S-1)

The intralayer symmetries are generated by {T1, T2, C2, C3}, where T1 : (x, y)→ (x+ 1, y), T2 : (x, y)→ (x, y + 1)are translations along the a1 and a2 directions, C2 : (x, y) → (y, x) a two-fold rotation, C3 : (x, y) → (−y, x − y) athree-fold counterclockwise rotation and I : (x, y) → (−x,−y) is an inversion. This symmetry group is equivalentto another group generated by {T1, T2, C2, C6}, where C6 ≡ C−1

3 I acts as C6 : (x, y) → (x − y, x). We sketch thesymmetry operations in Fig.S-1.

C2

T1

T2

C6

FIG. S-1. The symmetry operations.

These symmetry operations satisfy the following identities:

T−11 T2T1T

−12 = T−1

1 T−12 T1T2 = 1,

C−12 T1C2T

−12 = C−1

2 T2C2T−11 = 1,

C−16 T1C6T2 = C−1

6 T2C6T−12 T−1

1 = 1,

(C2)2 = (C6)6 = (C6C2)2 = 1.

(S-2)

We further require a time reversal symmetry T 2 = 1, satisfying

T−11 T T1T = T−1

2 T T2T = 1, C−12 T C2T = C−1

6 T C6T = 1. (S-3)

A2. k-space

The reciprocal lattice (Fig.S-2) k = k1b1 + k2b2 is studied in the dual basis b1 = 2π(1,√

33 ) and b2 = 2π(0, 2

√3

3 ).Here k1, k2 ∈ [0, 2π).

8

b1

b2

ΓK

M

FIG. S-2. The reciprocal lattice.

B. PSG ANALYSIS

We review the projective symmetry group (PSG) analysis in this section, following the discussion in [? ]. PSGis a powerful tool in the classification of symmetric spin liquids, which was first introduced in Ref.[28] and lists allpossible classes of lattice symmetry representations in the extended Hilbert space of spinons.

In the Nambu spinor representation Ψr = (fr↑, f†r↓, fr↓,−f

†r↑)

T , the mean field Hamiltonian is

HMF = −1

2

∑r,r′

(Ψ†rurr′Ψr′ + h.c.

), (S-4)

where u is the hopping matrix. The PSG method, in a word, seeks to classify gauge-inequivalent mean-field ansatzurr′ . In this Nambu representation, the spin operator Sr and the generator Gr of the SU(2) gauge transformationare given by,

Sr =1

4Ψ†r(σ ⊗ 12×2)Ψr, Gr =

1

4Ψ†r(12×2 ⊗ σ)Ψr. (S-5)

Under a symmetry operation O, Ψr transforms as

Ψr → UOGOO(r)ΨO(r) = GOO(r)UOΨO(r), (S-6)

where GOO(r) is the local gauge transformation and UO accounts for the rotation of the spin components due to thesymmetry operation O.

The commutation in the above equation can be understood from the observation [Sµr , Gνr ] = 0. From equation

(S-5), one further convinces himself of the block diagonal form of the gauge transformation GOr = 12×2 ⊗WOr , where

WOr is a 2× 2-dimensional matrix.The mean field ansatz (S-4) is invariant under the invariant gauge group (IGG = U(1)) with urr′ = G1†

r urr′GOO(r′).

To respect a lattice symmetry transformation O, the Hamiltonian should satisfy [30]

urr′ = GO†O(r)U†OuO(r)O(r′)UOGOO(r′). (S-7)

In the isotropic case with full SU(2) rotational symmetry, this matrix is trivial, i.e. U = 12×2. For YbMgGaO4, thissymmetry is partially broken and reduced to U(1), leading to a nontrivial U . (Note that the spin rotation symmetryU(1) is independent of the U(1) gauge symmetry, it is the latter that gives the name of U(1) spin liquid.)

A general group relation with the form O1O2O3O4 = 1 thus means the following constraint for the ansatz:

UO1GO1r UO2G

O2

O2O3O4(r)UO3GO3

O3O4(r)UO4GO4

O4(r) ∈ IGG. (S-8)

Using the commutation relation, we can rewrite it as

UO1UO2UO3UO4GO1r G

O2

O2O3O4(r)GO3

O3O4(r)GO4

O4(r) ∈ IGG. (S-9)

As the series of rotations O1O2O3O4 either rotate the spinons by 0 or 2π, UO1UO2UO3UO4

= ±14×4 always belongsto IGG. Thus the constraint further reduces to

GO1r G

O2

O2O3O4(r)GO3

O3O4(r)GO4

O4(r) ∈ IGG, (S-10)

9

or equivalently,

WO1r WO2

O2O3O4(r)WO3

O3O4(r)WO4

O4(r) ∈ {eiφσz | φ ∈ [0, 2π)}. (S-11)

For the space group symmetry of YbMgGaO4, the nontrivial spin rotation matrices are

UC2 = exp (−iπ2n2 · σ) =

(−eiπ/6

e−iπ/6

), UC6

= exp (iπ

3n3 · σ) =

(eiπ/3

e−iπ/3

), (S-12)

with n2 =(

12 ,√

32 , 0

)is the rotation axis for the C2 action and n3 = (0, 0, 1), since C6 ≡ C−1

3 I consists of a clockwise

rotation with respect to the z-axis by 2π/3.For the gauge part, we choose the canonical gauge [28] with IGG = {12×2 ⊗ eiφσz | φ ∈ [0, 2π)}. Then the gauge

transformation associated with the symmetry operation O takes the form

WOr = (iσx)nOeiφO[r]σz , (S-13)

where nO ∈ {0, 1}.In the case of translations, one can always choose a gauge so that

WT1r = (iσx)nT1 , WT2

r = (iσx)nT2 eiφT2[x,y]σz . (S-14)

The constraints (S-2) further demand n1 = n2 = 0 and φT2[x+1, y]−φT2

[x, y] ≡ φ1, φT2[x+1, y+1]−φT2

[x, y+1] ≡ φ2,with φ1 and φ2 to be determined. Here φ1 has the physical meaning of the flux through each unit cell of the triangularlattice. Since it is always possible to choose a gauge such that φT2

[0, y] = 0, we have φT2[x, y] = φ1x, thus φ2 = φ1.

Consequently, we obtain WT1r = 1, WT2

r = eixφ1σz .

For C6 with W C6r = (iσx)nC6 eiφC6

[x,y]σz , (S-11) leads to

χC6[r] = φ1xy − φ3x− φ4y − φ1

y(y − 1)

2, (S-15)

with φ3, φ4 undetermined. When nC6= 0, φ1 is not constrained, while for nC6

= 1, we further rquire φ1 = 0, π.

For C2 with WC2r = (iσx)nC2 eiφC2

[x,y]σz , (S-11) leads to φ1 = 0, π and φ3 + 2φ4 = 0 for both (nC2, nC6

) = (0, 0)and (0, 1), while for (nC2

, nC6) = (1, 0) and (1, 1), φ3=0. In all these four cases,{

φC2 = −xyφ1,

φC6= xyφ1 − φ1

y(y−1)2 .

(S-16)

C. THE SPECTRA

In this part we present the spectra of mean-field Hamiltonians U1A11 and U1A01.

C1. U1A11

In the U1A11 case, following the above section, one has φ1 = 0, (nC2 , nC6) = (1, 1), and

WT1= WT2

= 12×2, WC2= WC6

= WT = iσy. (S-17)

Combining (S-12) and (S-17), we obtain the transformation rules in Tab.I.One can then find the Hamiltonian that is invariant under these symmetry operations:

H =∑x,y

{t1 [ if†(x+1,y)↑f(x,y)↑ + if†(x,y+1)↑f(x,y)↑ − if†(x+1,y+1)↑f(x,y)↑

− if†(x+1,y)↓f(x,y)↓ − if†(x,y+1)↓f(x,y)↓ + if†(x+1,y+1)↓f(x,y)↓ ]

+ t′1 [ eiπ/3f†(x+1,y)↑f(x,y)↓ − f†(x,y+1)↑f(x,y)↓ + e2iπ/3f†(x+1,y+1)↑f(x,y)↓

+ e2iπ/3f†(x+1,y)↓f(x,y)↑ + f†(x,y+1)↓f(x,y)↑ + eiπ/3f†(x+1,y+1)↓f(x,y)↑ ]

+ t′2 [ eiπ/6f†(x+1,y−1)↑f(x,y)↓ + e5iπ/6f†(x+1,y+2)↑f(x,y)↓ − if†(x−2,y−1)↑f(x,y)↓

+ e5iπ/6f†(x+1,y−1)↓f(x,y)↑ − if†(x−2,y−1)↓f(x,y)↑ + eiπ/6f†(x+1,y+2)↓f(x,y)↑ ] + h.c.}

(S-18)

10

Symmetry Transformation Rules

T1

{f(x,y)↑ → f(x+1,y)↑

f(x,y)↓ → f(x+1,y)↓

T2

{f(x,y)↑ → f(x,y+1)↑

f(x,y)↓ → f(x,y+1)↓

C2

{f(x,y)↑ → eiπ/6f†(y,x)↑f(x,y)↓ → e−iπ/6f†(y,x)↓

C6

{f(x,y)↑ → eiπ/3f†(x−y,x)↓f(x,y)↓ → −e−iπ/3f†(x−y,x)↑

T

{f(x,y)↑ → f(x,y)↓f(x,y)↓ → −f(x,y)↑

TABLE I. U1A11 PSG analysis.

Here the t1 and t′1 terms describe the first-neighbor spin-preserving and spin-flipping hoppings, respectively. Thet′2 term denotes the second-neighbor spin-flipping hopping. Also note that spin-preserving next-nearest hopping isabsent, i.e., t2 = 0.

In k-space, we have

H(k1, k2) =

(εk ηkη∗k −εk

), (S-19)

where the spin-preserving and spin-flipping terms are

εk = t1 (sin k1 + sin k2 − sin(k1 + k2)) , (S-20)

ηk = t′1

(e5iπ/6 sin k1 − i sin k2 − eiπ/6 sin(k1 + k2)

)− t′2

(e2iπ/3 sin(k1 − k2)− eiπ/3 sin(k1 + 2k2)− sin(2k1 + k2)

).

(S-21)The resulting bands are Eν(k) = (−1)νE(k), with ν ∈ {1, 2} and

E(k) =√ε2k + |ηk|2. (S-22)

The two bands touch at Γ and M points. The function E(k) is plotted in Fig.S-3.

FIG. S-3. Spinon dispersion E2(k) in U1A11 state (the upper band). Blue (green) dots indicate location of saddle points(maxima) which show up as singularities in χ′′(ω) in Fig.1.

11

C2. U1A01

In the U1A01 case, φ1 = 0, (nC2 , nC6) = (0, 1), and the PSG analysis gives

WT1 = WT2 = WC2 = 12×2, WC6= WT = iσy. (S-23)

Combining with the spin rotation matrices, we obtain the following symmetry operations in Tab.II.

Symmetry Transformation Rules

T1

{f(x,y)↑ → f(x+1,y)↑

f(x,y)↓ → f(x+1,y)↓

T2

{f(x,y)↑ → f(x,y+1)↑

f(x,y)↓ → f(x,y+1)↓

C2

{f(x,y)↑ → −eiπ/6f†(y,x)↓f(x,y)↓ → e−iπ/6f†(y,x)↑

C6

{f(x,y)↑ → eiπ/3f†(x−y,x)↓f(x,y)↓ → −e−iπ/3f†(x−y,x)↑

T

{f(x,y)↑ → f(x,y)↓f(x,y)↓ → −f(x,y)↑

TABLE II. U1A01 PSG analysis.

The corresponding Hamiltonian invariant under such transformations is:

H =∑x,y

{t′1

[e5iπ/6f†(x+1,y)↑ − if

†(x,y+1)↑ + e−5iπ/6f†(x+1,y+1)↑

]f(x,y)↓

+ t′1

[eiπ/6f†(x+1,y)↓ − if

†(x,y+1)↓ + e−iπ/6f†(x+1,y+1)↓

]f(x,y)↑

+ t′2

[e−iπ/3f†(x+1,y−1)↑ + eiπ/3f†(x+1,y+2)↑ + f†(x+2,y+1)↑

]f(x,y)↓

+ t′2

[e−2iπ/3f†(x+1,y−1)↓ + e2πi/3f†(x+1,y+2)↓ − f

†(x+2,y+1)↓

]f(x,y)↑

+ t2

[−if†(x+1,y−1)↑ − if

†(x+1,y+2)↑ + if†(x+2,y+1)↑

]f(x,y)↑

+ t2

[if†(x+1,y−1)↓ + if†(x+1,y+2)↓ − if

†(x+2,y+1)↓

]f(x,y)↑ + h.c.

},

(S-24)

where again the t1,2 describe the spin-preserving nearest-neighbor and next-nearest-neighbor hoppings and t′1,2 describethe spin-flipping hoppings. Note that now t1 = 0, which most likely makes U1A01 state less energetically favorablethan U1A11 one. In k-space, we again have

H(k1, k2) =

(εk ηkη∗k −εk

), (S-25)

now with

εk =t2 [sin(k1 − k2) + sin(k1 + 2k2)− sin(2k1 + k2)] ,

−ηk =t′1

[e−2iπ/3 sin k1 + sin k2 + e−iπ/3 sin(k1 + k2)

]+ t′2

[eiπ/6 sin(k1 − k2) + e5iπ/6 sin(k1 + 2k2) + i sin(2k1 + k2)

].

(S-26)

The spectra E(k) is plotted in Fig.S-4. Spinon dispersions of the two U1A states along the path b1 + b2 in thereciprocal lattice, which contains Γ, K and M points, are compared in Figure S-5.

C3. With Magnetic Field

When an external magnetic field B = Bz z is present, the spectrum is shown in Fig.S-6.

12

FIG. S-4. Spinon dispersion E2(k) in U1A01 state (the upper band). Blue and red (green) dots indicate location of saddlepoints (maxima) which show up as singularities of χ′′(ω) in Fig. 2. The red saddle point, located between the K and M pointsof the Brillouin zone, is caused by the appearance of Dirac node at the K point in U1A01 state.

U1A11

U1A01

Γ K M K Γ

-2

-1

0

1

2

E(k)

FIG. S-5. Spinon dispersions E1,2(k) along the line Γ-K-M-K-Γ for U1A11 and U1A01 states.

FIG. S-6. Spinon dispersion E2(k) in U1A11 state in the presence of magnetic field along z, Bz = 1. Blue (green) dots indicatelocation of saddle points (maxima) which show up as singularities in χ′′(ω) in Fig. 3. Magnetic field along z breaks symmetrybetween the two maxima. The lower local maximum in the dispersion is causing discontinuity in χ′′(ω) at ω ≈ 4.2.

13

D. THE DYNAMICAL SUSCEPTIBILITY

Following equation (5) in the text, the imaginary part of dynamical susceptibility is

χ′′nn(ω) = − π

4N

∑k

δ(ω − 2E1(k)

E1(k)2

[ (ε2k + η′′2k

)sin2 θ cos2 φ+

(ε2k + η′2k

)sin2 θ sin2 φ+ |ηk|2 cos2 θ

+2η′kη′′k sin θ cosφ sinφ− 2εkη

′k sin θ cos θ cosφ+ 2εkη

′′k sin θ cos θ sinφ

], (S-27)

where η′k and η′′k denote the real and imaginary parts of ηk, respectively. The 2nd line of this equation is made ofoff-diagonal terms which were not shown in the main text because their contribution vanishes identically. This followsfrom the invariance of χ under symmetry transformations.

Symmetry ai bi ki

C2

{a1 → a2

a2 → a1

{b1 → b2

b2 → b1

{k1 → k2

k2 → k1

C3

{a1 → a2

a2 → −a1 − a2

{b1 → b2 − b1

b2 → −b1

{k1 → −k1 − k2k2 → k1

TABLE III. Transformation of coordinates under rotations.

Under the two- and three-fold rotations, the bases in real and reciprocal space transform as in Tab.III. χ transformdifferently for different mean-field Hamiltonians, due to their distinct dependences on k.

D1. U1A11

For the U1A11 Hamiltonian, using explicit form of εk and ηk we find that εk is invariant while ηk changes as:

C2 : εk → εk, η′k →1

2η′k +

√3

2η′′k, η′′k →

√3

2η′k −

1

2η′′k,

C3 : εk → εk, η′k → −1

2η′k −

√3

2η′′k, η′′k →

√3

2η′k −

1

2η′′k.

(S-28)

We start from the analysis of nxnz component of the susceptibility χ′′xz, which is proportional to∫dkεkη

′k. Since

the Brillouin zone is invariant under these C2 and C3 rotations, we only need to look at the integrand. From equation(S-28), one observes that εk is invariant under both C2 and C3, while η′k obtains different signs under the two actions.Consequently, these two symmetries constrain χ′′xz to be zero.

We now proceed to look at the nynz component, which is proportional to εkη′′k. A C3 rotation will lead to

η′′k →√

32 η′k − 1

2η′′k. But since the integration over εkη

′k vanishes, the only value for

∫dk εkη

′′k that preserves the

symmetry is zero.Finally, a similar argument leads to the vanishing of χ′′xy ∝

∫dkη′k η

′′k as well. These findings can be also verified

by numerical integration of the involved expressions over the Brillouin zone.

D2. U1A01

For U1A01 class, we instead have

C2 : εk → −εk, η′k → −1

2η′k −

√3

2η′′k, η′′k → −

√3

2η′k +

1

2η′′k,

C3 : εk → εk, η′k → −1

2η′k −

√3

2η′′k, η′′k →

√3

2η′k −

1

2η′′k.

(S-29)

Same arguments lead to the vanishing of off-diagonal terms of the dynamical susceptibility.

14

D3. Van Hove singularities

The peak in the middle of the absorption band in Fig.1, at ω ≈ 2.43, is caused by the van Hove singularity, of thesaddle point kind, of spinon dispersion E2(k) at k0 = (k0, 2π − k0) and also at (2π − k0, k0). Here k0 ≈ 1.97 andE2(k ≈ k0) ≈ 1.215 + 1.31(k1 + k2)2 − 0.23(k1 − k2)2. The saddle point region contributes to χ′′ as follows (hereu = k1 + k2, v = k1 − k2, we assume ω > 2E2(k0) and rescale u, v to obtain unit coefficients in front of u2, v2 terms)

χ′′(ω)SP ∼∫du

∫dv δ

(ω − 2E2(k0)− u2 + v2

)=

∫ Λ

−Λ

dv√ω − 2E2(k0) + v2

= ln( 4Λ2

ω − 2E2(k0)

)(S-30)

For ω < 2E2(k0) one first integrates over v, and then over u, to obtain the same result. We also used the fact that theprefactor of the delta-function in the expression for χ′′ reduces to a constant at k0. Therefore saddle point contributesχ′′(ω)SP ∼ ln(Λ2/|ω − 2E2(k0)|).

To understand behavior near the upper edge of the continuum we need to consider spinon spectrum near itsmaximum. For U1A11 the maximum occurs near K point where E2(k) = Emax − β(k2

1 + k1k2 + k22), where Emax =

3√

3/2 and β ≈ 0.38. We also need to know that near K point |ηk|2 → k21 + k1k2 + k2

2 while εk → const. As a resultχ′′zz (that is, θ = 0 part of susceptibility) and of χ′′xx (its θ = π/2 part) behave differently near ω = 2Emax.

Using k1 = r cosφ, k2 = r sinφ we obtain after simple manipulations

χ′′zz(ω) ∼∫dφ

∫ ∞0

r3dr δ

(r2 − 2Emax − ω

β(1 + cosφ sinφ)

)∼ (2Emax − ω)Θ(2Emax − ω) (S-31)

At the same time

χ′′xx(ω) ∼∫dφ

∫ ∞0

rdr δ

(r2 − 2Emax − ω

β(1 + cosφ sinφ)

)∼ Θ(2Emax − ω) (S-32)

reduces to a step-function. This explains behavior of ωχ′′ for different polarizations near ω = 2Emax = 3√

3 in Fig.1.Similar features are observed in Figures 2 and 3. Figure 3 features contributions from two different local maximathat spinon dispersion develops under magnetic field along z axis, see Fig.S-6. The smaller of these contributes todiscontinuity of χ′′ at ω ≈ 4.1, while the global maximum of the spectrum controls behavior near the upper boundary,at ω ≈ 6.2 in Figure 3.